Selecting the other side of the object in a boolean operation
When you are using Difference the other object is 'subtracted' from the one with the modifier. To get the other side you have to set the modifier to Intersect. This is the part where both objects overlap. The third option, Union would give you the boolean 'sum' of both objects.
In the image below the monkey head has the boolean modifier, the other object in the modifier is the cube.

BTW: notice the eye missing in the intersection case? The boolean modifier works on the object's volumes, but the monkey head is not a closed mesh, so the volume is not really defined. This may cause problems in some cases, so be aware of that.
EDIT: And as @stacker mentioned in the comments, since the boolean modifier works on the volume of the objects it might not work properly with a plane, at least when the edge of the plane is inside the volume of the other object.

UPDATE
There is no way to tell blender that it should take the other side of the object when using the boolean modifier set to Difference, since this is just the way the difference of two sets is defined: The difference of the sets A and B A\B are all points in A that are not in B. What points are in B is defined by the normal of your cutting plane; The normal points to the outside.
If you want the other half of the cube you have to flip the normal of the plane or set the boolean modifier to Intersect. The intersection of two sets A and B A^B is all points in A that are also in B. This is exactly the opposite of the difference.